QUESTIONS & SOLUTIONS OF AIEEE 2011

Transcription

1 QUESTIONS & SOLUTIONS OF AIEEE PART-I, PAPER- (MATHEMATICS & APTITUDE TEST) CODE Z Dat : -5- Duration : Hours Max. Marks : 9 IMPORTANT INSTRUCTIONS. Immdiatly fill th particulars on this pag of th Tst Booklt with Blu / Black Ball Point Pn. Us of pncil is strictly prohibitd.. This Tst Booklt consists of thr parts - Part-I, Part-II and Part-III. Part-I has objctiv typ qustions of Mathmatics consisting of FOUR (4) marks for ach corrct rspons. Mark your answrs for ths qustions in th appropriat spac against th numbr corrsponding in th appropriat spac against th numbr corrsponding to th qustion in th Answr Sht placd insid this Tst Booklt. Us Blu/Black Ball Point Pn only for writing particulars/marking rsponss on Sid- and Sid- of th Answr Sht. Part-III consists of qustions carrying 7 marks which ar to b attmptd on a sparat Drawing Sht which is also placd insid this Tst Booklt. Marks allottd to ach qustion ar writtn against ach qustion. Us colour pncil or crayons only on th Drawing Sht. Do not us watr colours. For ach incorrct rspons in Part-I and Part-II, on-fourth (¼) of th total marks allottd to th qustion would b dductd from th total scor. No dduction from th total scor, howvr, will b mad if no rspons is indicatd for an itm in th Answr Sht.. Thr is only on corrct rspons for ach qustion in Part-I and Part-II. Filling up mor than on rspons in ach qustion will b tratd as wrong rspons and marks for wrong rspons will b dductd accordingly as pr instructions abov. 4. Th tst is of hours duration. Th maximum marks ar On compltion of th tst, th candidats must hand ovr th Answr Sht of Mathmatics and Aptitud Tst-Part-I & II and th Drawing Sht of Aptitud Tst-Part-III to th Invigilator in th Room/Hall. Candidats ar allowd to tak away with thm th Tst Booklt of Aptitud Tst-Part-I & II. 6. Th CODE for this Booklt is Z. Mak sur that th CODE printd on Sid- of th Answr Sht and on th Drawing Sht (Part-III) is th sam as that on this booklt. Also tally th srial Numbr of th Tst Booklt, Answr Sht and Drawing Sht and nsur that thy ar sam. In cas of discrpancy in Cod or srial Numbr, th candidat should immdiatly rport th mattr to th Invigilator for rplacmnt of th Tst Booklt, Answr Sht and th Drawing Sht. Nam of th Candiat (in Capital lttrs) : Roll Numbr : in figurs : in words : Examination Cntr Numbr : Nam of Examination Cntr (in Capital lttrs) : Candidat's Signatur : Invigilator's Signatur :

2 PART-I (MATHEMATICS) a b c. If a plan mts th coordinat axs at A, B and C and ABC has cntroid at th point G,,, thn th quation of th plan is - () Sol. () x y z a b c Cntriod a () x y z a b c () x y z a b c (4) x y z a b c b Similarly c Equation of plan is - x y z + + c x y z + + a b c. Th latus rctum of th conic sction 9x +4y 6 is : - () 9 () /9 () /8 (4) 8/ 9x +4y 6 x y a 4 b 9 lngth of locus rctum is a b (4) 8. Th ara of th rgion boundd by th curvs y x, x + y + and x y is - () () / () 7/ (4) 8/ Sol. () Rquird Ara ( x ) dx + x x tan9 tan7 tan6 + tan8 is qual to : () () 4 () (4) Sol. () tan9 tan7 cot7 + cot9 sin9 cos9 sin7 cos 7 cos9 sin9 cos 7 sin7 Pag #

3 sin9cos9 sin7cos 7 sin 8 sin ( 5 ) 5 () 4 Ans. ( 5 ) 5 5. Statmnt - : Th quation x + y rprsnts a paralllogram. Statmnt - : Lins x + y and x + y ar paralll. Also lins x y and x + y ar paralll. () Statmnt - is tru, Statmnt - is fals. () Statmnt - is fals, Statmnt - is tru. () Statmnt - is tru, Statmnt - is tru ; Statmnt - is corrct xplanation for Statmnt -. (4) Statmnt - is tru, Statmnt - is tru ; Statmnt - is not a corrct xplanation for Statmnt -. x + y rprsnt paralllogram lins x + y, x + y ar paralll also lins x y, x + y ar also paralll / 6. min (sinx, cosx) dx, qual to : () + () () (4) / 4 / sin x dx + cos x dx / 4 / 4 / / cos x + sin x + Pag #

4 7. Lt OA a, OB b a and OC b whr O is th origin. If p is th ara of th quadrilatral OABC and q is th ara of th paralllogram with OA and OC as adjacnt sids thn p is qual to - () 6 p () q 6 () 6q (4) q/6 Sol. () Ara of paralllogram with OA and OC as adjacnt sids q a b Ara of quadrilatral OABC is Ara (OAB) + Ara of (OBC) ( OA OB) (OB OC) a (b a) + a b + 5 a b p 6 p 6q a b (b a) b OA OC 8. If x, x, x,..., x ar in A.P. thn th valu of x x 4 x 7 x 4 x 7 x x 7 x x is - () 9 () 7 () (4) Sol. () x, x, x,..., x ar in A.P. x a x a + d x a + d x a + Now x x 4 x 7 x 4 x 7 x x 7 x x a ad a6d ad a6d a9d a6d a9d ad a. a. a d 6d d 6d 9d 6d 9d d (multiply c by d thn c and c ar idntical w gt zro) Pag # 4

5 9. Th valu of and such that lim x x x x ar : (), /4 (), /4 (), 5/4 (4), 5/4 Sol. () lim x ( x ) x(x ) (x ) x lim x ( )x Limit will b xist ( )x ( ) x and, 4 5. Th function f(x) x x has : () a maximum at x () nithr maximum nor minimum at x () a minimum at x (4) a maximum at x Sol. () f(x) x x f '(x). x x x x ( x) (x ) x for maxima and minima at x, it has minimum valu. Ara of a triangl with vrtics givn by z, iz, z + iz, whr z is any complx numbr is : () z () () z (4) z Ara of Rctangl OABC is z. z z. Th acut angl btwn th tangnts drawn from th point (, 4) to th parabola y 4x is : () /4 () /6 () / (4) / Pag # 5

6 Parabola y 4x tangnt quation y mx + m pass (, 4) 4 m + m m 4m + m + m 4 m m Angl btwn tangnts tan m m m m (4) 4() tan /. A committ consisting of at last thr mmbrs is to b formd from a group of 6 boys and 6 girls such that it always has a boy and a gir. Th numbr of ways to form such committ is - () 7 5 () 7 () 6 5 (4) 7 5 Numbr of ways to slct atlast thr passons Total no slction prson slctd prson slctd C 79 Rquird ways 79 [ 6 C + 6 C + 6 C C C C 6 ] 79 ( 7 44) 7 5 x, 4. If f(x) x, x x thn th drivativ of f(x) at x is - Sol. () () () () (4) x, f(x) x, x x at x, it is continuous f '(x) x,, x x f '(x) Ans. 5. Shortst distanc btwn z-axis and th lin x y 5 z 5 is - () Sol. () () () (4) Equation of z-axis x y z b kˆ and lin x y 5 z 5 Pag # 6

7 d î + ĵ 5 kˆ b d ĵ î (c a).(b d) S.D b d ( î 5ĵ k).( î ĵ) 6. Statmnt - : ~ (A ~B) is quivalnt to A B. Statmnt - : A v (~(A^~B)) a tautology. () Statmnt - is tru, Statmnt - is fals. () Statmnt - is fals, Statmnt - is tru. () Statmnt - is tru, Statmnt - is tru ; Statmnt - is a corrct xplanation for Statmnt -. (4) Statmnt - is tru, Statmnt - is tru ; Statmnt - is not a corrct xplanation for Statmnt -. Statmnt - : A B ~B A B A B ~ (A B) T F T F T F F T F F T F T T F T F T F F T T F T so statmnt - is tru Statmnt - A v (~(A^ ~B)) A v (~A v B) (A v ~ A) v B t v B t Statmnt () is tru. 7. Statmnt - : Th function f(x) x x sin x is vn. Statmnt - : Product of two odd functions is an vn function. () Statmnt - is tru, Statmnt - is fals. () Statmnt - is fals, Statmnt - is tru. () Statmnt - is tru, Statmnt - is tru ; Statmnt - is a corrct xplanation for Statmnt -. (4) Statmnt - is tru, Statmnt - is tru ; Statmnt - is not a corrct xplanation for Statmnt -. Sol. () Statmnt - : f(x) x x sin x f( x) ( x) x sin x f( x) x x sin x f(x) vn function, statmnt- is tru. Statmnt - : lt f(x) and g(x) ar two odd functions and f(x) f '(x) g(x) h( x) f( x) g( x) h( x) f(x) g(x) f(x) Statmnt - is also tru. but it dos not xplain statmnt - 8. If th man and standard dviation of obsrvations X, X, X,..., X ar 5 and rspctivly, thn i i X is qual to : () 6 () 5 () 5 (4) 5 Pag # 7

8 X 5 S.D. (S.D.) X i 9 (X) X i 5 X 5 i 9. If th sum of th cofficints in th xpansion of (x+y) n is 48, thn th gratst cofficint in th xpansion is : () C 6 () C 6 () C 6 (4) C 7 Sol. () (x+y) n nc x n + nc x n- y nc y n put x y n C + C + C C n 48 n n gratst coff. in th xpansion of (x+y) is C x t. Lt f : R R b dfind as f(x) t dt. Thn th curv y f(x) is - () a hyprbola () an llips () a straight lin (4) a parabola x t f(x) t dt. x t 4 4 dt t t t x n( t) t t 4n( t) x (n) 5 + 4n f(x) x ln 5 + 4n y This is quaiton of parabola. Lt p, q, r b ral numbrs such that p + q + r. Th systm of linar quations x + y z p x + 6y z q x y + 7z r has at last on solution if : () 5p + q r () 5p q r () 5p + q + r (4) 5p q + r Sol. () x + y z p x + 6y z q x y + 7z r Pag # 8

9 D 6 7 (4 ) (4+) ( 4 6) 5 + D p q r 6 7 p() (7q + r) ( q 6r) p 4q r + 6q + 8r p 8q 4r 4(5p q r) If D, thn thr ar infinit solutions which confirm at last on solution. 5p q r. Th quation of a curv is givn by y f(x). whr f ''(x) is a continuous function. Th tangnts at points (, f(), (, f()) and (, f()) mak angls 6, and 4 rspctivly with th positiv x-axis. Thn Sol. () f'(x) f''(x)dx + f''(x) dx is : () () (*) f'(x) f''(x)dx... (I ) + f''(x) dx... (I ) (4) I f '(x) f'(x) f''(x)dx I ( f'()) (f'()) I I ( ) I I f '(x) f '() f '() I I + I +. Considr th folowing rlations R { (x, y) : x, y ar intgrs and x ay or y ax for som intgr a } R {(x, y) : x, y ar intgrs and ax + by for som intgrs a, b} Thn () R is an quivalnc rlation but R is not () R, R ar not quivalnc rlations. () R, R ar quivalnc rlations. (4) R is an quivalnc rlation but R is not Sol. () Rlation R : (i) x ax for a rflxiv (ii) x ay Pag # 9

10 y a x a may not b intgr not symmtric so R is not quivalanc Rlation R : ax + ax ax ax not possibl so R is not rflxiv so not quivalanc 4. If x x + is factor of x 4 ax + b, thn th quation whos roots ar a and b is : () x + 9x () x + 9x + () x 9x (4) x 9x + x x + is factor of x 4 ax + b but f(x) x 4 ax + b x x + is a factor of f(x) f() a + b a b... (i) also f() 6 4a + b 4a b 6... (ii) (ii) (i) a 5 a 5 b 4 x 9x + is th dsivd quaiton 5. If P(A).4, P(B').6 and P(AB).5, thn th valu of P(A A'B') is () 7 P(A).4 P(A').6 P(A/A'B') () 7 P(A (A' B' )) P(A' B' ) () 7 4 (4) 7 5 P(A) P(A B) P(A B) Statmnt - : If f(x) (x )(x ), thn Roll's thorm is applicabl to f(x) in th intrval [, ] Statmnt - : Man valu thorm is applicabl to f(x) (x )(x ) in th intrval [, 4] () Statmnt - is tru, Statmnt - is fals. () Statmnt - is fals, Statmnt - is tru. () Statmnt - is tru, Statmnt - is tru ; Statmnt - is corrct xplanation for Statmnt -. (4) Statmnt - is tru, Statmnt - is tru ; Statmnt - is not a corrct xplanation for Statmnt -. f(x) (x )(x ) is continuous and diffrntiabl x R. so man valu thorm is applicabl. Also f() f(). So roll's thorm is also applicabl. 7. If a and b ar such that a sin bcos for < /4, thn () sin cos a b + a b a b a b a a () b cos b a a b + a b a b a b () cos (4*) quals : cos cos tan cos cos Pag #

11 dy 8. If y(x) is a solution of th diffrntial quation +y, thn dx lim y(x) is qual to - x (*) Sol. () dy dy +y dx y () () (4) dx n( y) x + c n( y) x c y x. c y x c. lim y x lim x x c. 9. Lt A and B b two matrics. Statmnt - : A(adj A) A I Statmnt - : adj(ab) (adj A) (adj B) () Statmnt - is tru, Statmnt - is fals. () Statmnt - is fals, Statmnt - is tru. () Statmnt - is tru, Statmnt - is tru ; Statmnt - is corrct xplanation for Statmnt -. (4) Statmnt - is tru, Statmnt - is tru ; Statmnt - is not a corrct xplanation for Statmnt -. Sol. () adj(ab) (adj B) (adj A) so statmnt is fals. Th quation of a circl of ara squar units for which ach of th two lins x + y and x y 5 is diamtr is : () x + y + x + 8y 5 () x + y x + 8y 5 () x + y x 8y 5 (4) x + y + x 8y 5 r r solving x + y and x y 5 w gt cntr of circl as (, 4) So quation of circl is (x + ) + (y 4) x + y + x 8y 5 Pag #

12 PART-II (APTITUDE) Dirctions (For Q. to 5) Th -D problm figur shows th viw of an objct. Idntify its corrct top viw from amongst th answr figurs. Problm Answr Figurs. () () () (4) Ans.. () () () (4) Ans.. () () () (4) Ans. 4. () () () (4) Ans. 5. () () () (4) Pag #

13 Dirctions (For Q.6 to 8) Which on of th answr figurs is th corrct mirror imag of th problm figur with rspct to X X? Problm Answr Figurs 6. () () () (4) 7. () () () (4) Ans. 8. () () () (4) Ans. Dirctions (For Q.9 to 4) Which on of th answr figurs will complt th squnc of th thr problm figurs? Problm Answr Figurs 9. () () () (4) Ans. 4. () () () (4) Ans. 4. () () () (4) Pag #

14 Dirctions (For Q.4 to 48) Th -D problm figur shows th viw of an objct. Idntify its corrct front viw, from amongst th answr figurs, looking in th dirction of arrow. Problm Answr Figurs 4. () () () (4) Ans. 4. () () () (4) Ans. 44. () () () (4) 45. () () () (4) Ans. Pag # 4

15 46. () () () (4) 47. () () () (4) Ans. 48. () () () (4) Ans. Dirctions (For Q.49 to 5) Find out th total numbr of surfacs of th objct givn blow in th problm figur. Problm 49. () 8 () () 7 (4) 9 5. () 4 () 5 () (4) Ans. Pag # 5

16 5. () () 4 () (4) Ans. 5. () 5 () 6 () 4 (4) 5. () () () 5 (4) 4 Ans. Dirctions (For Q.54 to 55) Find th odd figurs out in th problm figurs givn blow : 54. () () () (4) Ans. 55. () () () (4) Pag # 6

17 Dirctions (For Q.56 to 57) How many total numbr of triangls ar thr in th problm figur givn blow? Problm Figur 56. () 8 () 7 () 4 (4) () 6 () () 5 (4) 4 Ans. Dirctions (For Q.58 to 6) Which on of th answr figurs shows th corrct viw of th -D problm figur, aftr th problm figur is opnd up? Problm Figur Answr Figur 58. () () () (4) Ans. 59. () () () (4) Ans. Pag # 7

18 6. () () () (4) Ans. Dirctions (For Q.6) How many total numbr of squars ar thr in th problm figur givn blow? 6. () 9 () () (4) 6 Dirctions (For Q.6 to 6) Th problm figur shows th top viw of an objct. Idntify its corrct front viw, from amongst th snwr figurs. 6. () () () (4) Ans. 6. () () () (4) Ans. Dirctions (For Q.64 to 65) On of th following answr figurs is hiddn in th problm figur, in th sam siz and dirction. Slct, which on is corrct? 64. () () () (4) Ans. Pag # 8

19 65. () () () (4) Ans. 66. Who was th architct of Sansad Bhawan, Nw Dlhi? () A.P. Kanvind () Alvar Alto () Charls Corra (4) Hrbrt Bakr 67. Which on of th following is an Earthquak rsistant structur? () Random ston masonary. () Mud walls. () RCC framd. (4) Load baring brick walld. Ans. 68. What is Grn Architctur? () Whr maximum grn plants ar () Whr grn colourd glass is usd. () Whr buildings ar paintd grn. (4) Whr building matrial usd hav consumd last nrgy. 69. Which structural componnt can b built without stl rinforcmnt? () Doms () Foldd concrt roof slabs () Concrt bams (4) Flat concrt roof slabs Ans. 7. In buildings, which parts ar rsponsibl for maximum hat Eain? Ans. () Windows () Chajjas () Roofs (4) Walls 7. Trs should b plantd on which sid to protct buildings from hat gain? Ans. () Wst sid () East sid () North (4) North-East sid 7. Cavity walls ar bst suitd against: Ans. () Rain () Hat () Dust (4) Light Pag # 9

20 7. Who amongst th following is an architct? Ans. () Usha Uthup () Raj Rwal () Ravi Shankar (4) Shankr Mahadvan 74. Th Dom of influncd by: Ans. () Stupa () Gurdwara () Mosqu (4) Tmpl 75. Which on of th following is not a matching st? () Hat - Insulation () Stl - Mud () Concrt - Bam (4) Sound - Vibration Ans. 76. Nam th city, whr canals ar usd for transportation channls: Ans. () Vnic () Canbrra () Tokyo (4) Manhattan 77. Tracing papr is: Ans. () Opaqu () Black () Transparnt (4) Smi-transparnt 78. Nw Gugnhium musum is dsignd by: Ans. () LM. Pi () Frank O Ghry () F.L. Wright (4) Charls Corra 79. Horizontal sun shads ar rquird to protct windows on which facad of a building? Ans. () North () South () East (4) Wst 8. Which is not a sound absorbing matrial? Ans. () Ground Glass () Jut Bags () Thrmocol (4) Glass Pag #